Math

CalculusMultivariable CalculusDamped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation d 2 y d t 2 + 2 λ d y d t + ω 2 y = 0 The zeros of its characteristic equation are m 1 = − λ + λ 2 − ω 2 and m 1 = − λ − λ 2 − ω 2 For λ 2 − ω 2 > 0 , the system is overdamped; λ 2 − ω 2 = 0 , it is critically damped; and for λ 2 − ω 2 < 0 , it is underdamped. Determine whether the differential equation represents an overdamped, critically damped, or underdamped system. Find the particular solution that satisfies the initial conditions. Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system. d 2 y d t 2 + 8 d y d t + 16 y = 0 y ( 0 ) = 1 and y ′ ( 0 ) = 1BuyFind*launch*

11th Edition

Ron Larson + 1 other

Publisher: Cengage Learning

ISBN: 9781337275378

Chapter 16, Problem 11PS

Textbook Problem

Damped Motion In Exercises 11-14, consider a damped mass-spring system whose motion is described by the differential equation

The zeros of its characteristic equation are

and

For

Determine whether the differential equation represents an overdamped, critically damped, or underdamped system.

Find the particular solution that satisfies the initial conditions.

Use a graphing utility to graph the particular solution found in part (b). Explain how the graph illustrates the type of damping in the system.

Multivariable Calculus

Ch. 16.1 - Exactness What does it mean for the...Ch. 16.1 - Integrating Factor When is it beneficial to use an...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Testing for Exactness In Exercises 3-6, determine...Ch. 16.1 - Solving an Exact Differential Equation InExercises...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential Equation In...

Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Solving an Exact Differential Equation In...Ch. 16.1 - Solving an Exact Differential EquationIn Exercises...Ch. 16.1 - Graphical and Analytic Analysis In Exercises 15...Ch. 16.1 - Graphical and Analytic AnalysisIn Exercises 15 and...Ch. 16.1 - Finding a Particular Solution In Exercises17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding a Particular Solution In Exercises 17-22,...Ch. 16.1 - Finding a Particular SolutionIn Exercises 17-22,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Finding an Integrating Factor In Exercises 23-32,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Using an Integrating Factor In Exercises 33-36,...Ch. 16.1 - Integrating Factor Show that each expression is...Ch. 16.1 - Integrating FactorShow that thedifferential...Ch. 16.1 - Tangent Curves In Exercises 39-42, use agraphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Tangent Curves In Exercises 39-42, use a graphing...Ch. 16.1 - Finding an Equation of a Curve In Exercise 43 and...Ch. 16.1 - Finding an Equation of a Curve In Exercises 43 and...Ch. 16.1 - Cost In a manufacturing process where y=C(x)...Ch. 16.1 - HOW DO YOU SEE? The graph shows several...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - Eulers Method In Exercises 47 and 48, (a) use...Ch. 16.1 - EXPLORING CONCEPTS Exact Differential Equation In...Ch. 16.1 - EXPLORING CONCEPTS Exact Differential Equation In...Ch. 16.1 - Exact Differential Equation Find all nonzero...Ch. 16.1 - Exact Differential Equation Find all nonzero...Ch. 16.1 - Every separable equation is an exact equation.Ch. 16.1 - Every exact equation is a separable equation.Ch. 16.1 - If Mdx+Ndy=0 is exact, then...Ch. 16.1 - If Mdx+Ndy=0 is exact, then xMdx+xNdy=0 is also...Ch. 16.2 - Linear Differential Equation Determine the order...Ch. 16.2 - Linearly Independent Describe what is means for...Ch. 16.2 - Using Zeros The zeros of the characteristics...Ch. 16.2 - Finding a General Solution Explain how to find the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Verifying a SolutionIn Exercises 5-8, verify the...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a General Solution In exercises 9-36, find...Ch. 16.2 - Finding a particular Solution Consider the...Ch. 16.2 - Finding a Particular Solution Determine C and ...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Find a Particular Solution: Initial ConditionsIn...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding a Particular Solution: Boundary...Ch. 16.2 - Finding another Solution Show that the equation...Ch. 16.2 - General Solution of a Differential Equation What...Ch. 16.2 - Several shock absorbers are shown at the right. Do...Ch. 16.2 - Give a geometric argument to explain why the graph...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a Spring In Exercise 55-58, match the...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Motion of a SpringIn Exercise 59-64, a 32-pound...Ch. 16.2 - Real Zeros The characteristics equation of the...Ch. 16.2 - Complex Zero The characteristics equation of the...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - True or False? In exercises 67-70, determine...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.2 - Wronskian The Wronskian of two differentiable...Ch. 16.3 - Writing What is the form of the general solution...Ch. 16.3 - Choosing a MethodDetermine whether you woulduse...Ch. 16.3 - Method of Undetermined Coefficients InExercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Method of Undetermined Coefficients In Exercises...Ch. 16.3 - Choosing the Form of the ParticularSolution In...Ch. 16.3 - Choosing the Form of the ParticularSolution In...Ch. 16.3 - Choosing the Form of the Particular Solution In...Ch. 16.3 - Choosing the Form of the Particular Solution In...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Method of Undetermined CoefficientsIn Exercises...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Using Initial Conditions In Exercises 17-22, solve...Ch. 16.3 - Method of Variation of Parameters InExercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Method of Variation of Parameters In Exercises...Ch. 16.3 - Electrical Circuits In Exercises 29 and 30, use...Ch. 16.3 - Electrical Circuits In Exercises 29 and 30, use...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - Motion of a Spring In Exercises 31-34, use the...Ch. 16.3 - EXPLORING CONCEPTS Motion of a Spring Rewrite yh...Ch. 16.3 - HOW DO YOU SEE IT? The figure shows the particular...Ch. 16.3 - Motion of a SpringRefer to the differential...Ch. 16.3 - Solving a Differential Equation Solve the...Ch. 16.3 - True or False? In Exercises 39 and 40, determine...Ch. 16.3 - True or False? In Exercises 39 and 40, determine...Ch. 16.3 - PUTNAM EXAM CHALLENGE For all real x, the...Ch. 16.4 - CONCEPT CHECK Power Series Solution Method...Ch. 16.4 - CONCEPT CHECK Recursion Formula What is a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Power Series Solution In Exercises 3-6, use a...Ch. 16.4 - Finding Terms of a Power Series SolutionIn...Ch. 16.4 - Finding Terms of a Power Series SolutionIn...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - Approximation by Taylor Series In Exercises 9-14,...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - EXPLORING CONCEPTS Using Different MethodsIn...Ch. 16.4 - Investigation Consider the differential equation...Ch. 16.4 - HOW DO YOU SEE IT? Consider the differential...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Verifying that a Series Converges In Exercises...Ch. 16.4 - Airys Equation Find the first six terms of the...Ch. 16 - Testing for Exactness In Exercises 1 and 2,...Ch. 16 - Testing for Exactness In Exercises 1 and 2,...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Solving an Exact Differential Equation In...Ch. 16 - Graphical and Analytical Analysis In Exercise 7...Ch. 16 - Graphical and Analytical Analysis In Exercise 7...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding a Particular Solution In Exercises 9-12,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Finding an Integrating Factor In Exercises 13-16,...Ch. 16 - Verifying a Solution In Exercises 17-18, verify...Ch. 16 - Verifying a Solution In Exercises 17 and 18,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a General Solution In exercises 19-28,...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Initial Conditions...Ch. 16 - Finding a Particular Solution: Boundary...Ch. 16 - Finding a Particular Solution: Boundary...Ch. 16 - Motion of a SpringIn Exercise 35-36, a 64-pound...Ch. 16 - Motion of a SpringIn Exercise 35-36, a 64-pound...Ch. 16 - Method of Undetermined Coefficients In...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Choosing the Form of the Particular Solution...Ch. 16 - Choosing the Form of the Particular Solution In...Ch. 16 - Method of Undetermined Coefficients In Exercises43...Ch. 16 - Method of Undetermined Coefficients In Exercises...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Using Initial Conditions In Exercises 45-50, solve...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Method of Variation of Parameters In Exercises...Ch. 16 - Electrical Circuit The differential equation...Ch. 16 - Investigation The differential equation...Ch. 16 - Think About It (a) Explain how, by observation,...Ch. 16 - Think About It Explain how you can find a...Ch. 16 - Power Series Solution In Exercises 59 and 60, use...Ch. 16 - Power Series Solution In Exercises 59 and 60, use...Ch. 16 - Approximation by Taylor Series In Exercises 61 and...Ch. 16 - Approximation by Taylor Series In Exercises 61 and...Ch. 16 - Finding a General Solution Find the value of k...Ch. 16 - Using an Integrating Factor The differential...Ch. 16 - Finding a General Solution Find the general...Ch. 16 - Finding a General Solution Find the general...Ch. 16 - Distinct Real Zeros Given that the characteristic...Ch. 16 - Limit of a Solution Given that a and b are...Ch. 16 - Trivial and Nontrivial Solutions Consider the...Ch. 16 - Eulers Differential Equation Eulers differential...Ch. 16 - Pendulum Consider a pendulum of length L that...Ch. 16 - Deflection of a Beam A horizontal beam with a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Damped Motion In Exercises 11-14, consider a...Ch. 16 - Airys Equation Consider Airys equation given in...Ch. 16 - ChebyshevsEquation ConsiderChebyshevs equation...Ch. 16 - Bessels Equation: Order Zero The differential...Ch. 16 - Bessels Equation: Order One The differential...Ch. 16 - Hermites Equation Consider Hermites Equation...Ch. 16 - Laguerres Equation Consider Laguerres Equation...

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